Only 40000 horses? That number sounds a bit low. Can we get more details of the calculation?
Ugh. Sorry I worded that so poorly. I meant horses for 40,000 soldiers. If you assume each one has four horses, that's 160,000 horses as a conservative estimate. As for calculations, I went and took a look for the original document, but it doesn't seem to be available anymore.
European geography is tricky, but China also has high mountains and big rivers, and it didn't help them. The Volga is a bigger river than any other in Europe, and the Mongols crossed it.
Castles are quite some defense, but the Mongols managed to take Chinese and Islamic fortresses, which are on a higher technological level. And the Mongols have gunpowder.
The issue with terrain - and the Carpathians in particular - is that the basin forms a double chokepoint. The northern route goes through Bohemia - heavily mountainous and forested, not horse archer country for sure. The southern route has better grazing and open terrain. The Hungarian Plain is down this way, and the Mongols approached this direction OTL. The issue here is that the only way west leads through a narrow pass directly across Vienna - and this route has been continuously fortified for centuries by the 13th century because of all the invaders that have come this way in generations past. It's not that the terrain alone will stop you, it's that it pretty much forces you to confront the heaviest defenses head on.
Higher technology doesn't necessarily make for better castle construction. And if you look closely at Mongol history, you'll notice that the most heavily defended cities and fortresses either took them a very long time to besiege or else were intimidated into surrendering. It seems silly to assume they'd always be that fortunate.
As for gunpowder, remember that the Mongols invaded Europe in the 1240's. Gunpowder didn't make for effective siege weapons until later. You certainly don't hear about Mongol cannons much, do you?
The real problem here is that the castles and terrain will slow the Mongols down considerably, and once their armies are stalled, the horses will die off from lack of grazing. It's a pretty common circumstance in siege warfare, and a big part of the reason why all-cavalry armies have never stayed in Europe for long, historically.
And about knights: In the battle of Liegnitz/Legnica, the Teutonic knights and their allies were trounced by the Mongols. The Mongols used their usual tactic: A fake retreat, the knights attack, throw caution in the wind, also the majority of the Mongols are hidden behind a smoke screen... the knights were slaughtered.
Never said that Mongols couldn't beat knights, just that they wouldn't always do so. If Patay had never happened, would you assume that longbowmen were invincible against knights? I'd think Ain Jalut or Lechfeld would be good enough examples of what happens when your light cavalry army gets engaged by heavier troops in close combat. My larger point being that tactical success comes and goes, and that if your army can't cope with the consequences, then you can't count on winning every time.
If anything, the Mongols may ignore Europe because it was too poor. The silver mines of Tyrol aren't open yet, Europe doesn't have silk like China...
By the 1200's that was less true than you'd think. The wool trade was really starting to come into its own at this point, enriching cities like Antwerp and Bruges. And the Italian cities were loaded, of course. France also experienced an immense population boom in this period. Really, the 13th century was the time Europe stopped being a backwater and started really pulling closer to the Middle East.
